The Jarník-Besicovitch Theorem for Geometrically Finite Kleinian Groups

Richard Hill; Sanju Velani

doi:10.1112/S0024611598000550

The Jarník-Besicovitch Theorem for Geometrically Finite Kleinian Groups

Richard Hill, Sanju Velani

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let G be a geometrically finite Kleinian group with parabolic elements and let p be any parabolic fixed point of G. For each positive real t, let Wp t denote the set of limit points of G for which the inequality $| x-g(p) |\leq |g^\prime(0)|^{\tau}$ is satisfied for infinitely many elements g in G. This subset of the limit set is precisely the analogue of the set of t-well approximable numbers in the classical theory of metric Diophantine approximation. In this paper we consider the following question. What is the ‘size’ of the set Wp t expressed in terms of its Hausdorff dimension? We provide a complete answer, namely that for t = 1, $\dim {\cal W}_{p} (\tau) = \min \left\{ \frac{\delta + \mbox{rk}(p) (\tau - 1) }{2 \tau - 1}, \, \frac{\delta}{\tau}\right\},$ where rk (p) denotes the rank of the parabolic fixed point p. 1991

Original language	English
Pages (from-to)	524-550
Number of pages	27
Journal	Proc. Lond. Math. Soc
Volume	77
Issue number	3
DOIs	https://doi.org/10.1112/S0024611598000550
Publication status	Published - Nov 1998

Keywords

Diophantine approximation
Kleinian groups
Hausdorff dimension
geodesic excursions

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10.1112/S0024611598000550

Cite this

@article{9af0c35ad2a64ee699757899ec730120,

title = "The Jarn{\'i}k-Besicovitch Theorem for Geometrically Finite Kleinian Groups",

abstract = "Let G be a geometrically finite Kleinian group with parabolic elements and let p be any parabolic fixed point of G. For each positive real t, let Wp t denote the set of limit points of G for which the inequality $| x-g(p) |\leq |g^\prime(0)|^{\tau}$ is satisfied for infinitely many elements g in G. This subset of the limit set is precisely the analogue of the set of t-well approximable numbers in the classical theory of metric Diophantine approximation. In this paper we consider the following question. What is the {\textquoteleft}size{\textquoteright} of the set Wp t expressed in terms of its Hausdorff dimension? We provide a complete answer, namely that for t = 1, $\dim {\cal W}_{p} (\tau) = \min \left\{ \frac{\delta + \mbox{rk}(p) (\tau - 1) }{2 \tau - 1}, \, \frac{\delta}{\tau}\right\},$ where rk (p) denotes the rank of the parabolic fixed point p. 1991 ",

keywords = "Diophantine approximation, Kleinian groups, Hausdorff dimension, geodesic excursions",

author = "Richard Hill and Sanju Velani",

year = "1998",

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language = "English",

volume = "77",

pages = "524--550",

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TY - JOUR

T1 - The Jarník-Besicovitch Theorem for Geometrically Finite Kleinian Groups

AU - Hill, Richard

AU - Velani, Sanju

PY - 1998/11

Y1 - 1998/11

N2 - Let G be a geometrically finite Kleinian group with parabolic elements and let p be any parabolic fixed point of G. For each positive real t, let Wp t denote the set of limit points of G for which the inequality $| x-g(p) |\leq |g^\prime(0)|^{\tau}$ is satisfied for infinitely many elements g in G. This subset of the limit set is precisely the analogue of the set of t-well approximable numbers in the classical theory of metric Diophantine approximation. In this paper we consider the following question. What is the ‘size’ of the set Wp t expressed in terms of its Hausdorff dimension? We provide a complete answer, namely that for t = 1, $\dim {\cal W}_{p} (\tau) = \min \left\{ \frac{\delta + \mbox{rk}(p) (\tau - 1) }{2 \tau - 1}, \, \frac{\delta}{\tau}\right\},$ where rk (p) denotes the rank of the parabolic fixed point p. 1991

AB - Let G be a geometrically finite Kleinian group with parabolic elements and let p be any parabolic fixed point of G. For each positive real t, let Wp t denote the set of limit points of G for which the inequality $| x-g(p) |\leq |g^\prime(0)|^{\tau}$ is satisfied for infinitely many elements g in G. This subset of the limit set is precisely the analogue of the set of t-well approximable numbers in the classical theory of metric Diophantine approximation. In this paper we consider the following question. What is the ‘size’ of the set Wp t expressed in terms of its Hausdorff dimension? We provide a complete answer, namely that for t = 1, $\dim {\cal W}_{p} (\tau) = \min \left\{ \frac{\delta + \mbox{rk}(p) (\tau - 1) }{2 \tau - 1}, \, \frac{\delta}{\tau}\right\},$ where rk (p) denotes the rank of the parabolic fixed point p. 1991

KW - Diophantine approximation

KW - Kleinian groups

KW - Hausdorff dimension

KW - geodesic excursions

U2 - 10.1112/S0024611598000550

DO - 10.1112/S0024611598000550

M3 - Article

VL - 77

SP - 524

EP - 550

JO - Proc. Lond. Math. Soc

JF - Proc. Lond. Math. Soc

IS - 3

ER -